Nonlinear Wave Motion

非线性波动

• 批准号: 2306290
• 负责人: Mark Ablowitz
• 金额: $ 20万
• 依托单位: University of Colorado at Boulder
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-09-01 至 2026-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2306290&HistoricalAwards=false
• 关键词: Nonlinear; Wave; Motion

Wave motion is fundamental in nature: Light waves impact the way people see, sound waves allow people to hear, and water waves are critical to the motion of ships and tidal energy. This project is focused on nonlinear wave propagation, that is the study of large amplitude or high-power signals, and aims to enhance the ability of scientists to develop insight and understanding of large amplitude waves and of the associated dynamics of wave energy transfer. The results of the investigation will benefit applications of nonlinear wave motion, which include tsunamis, rogue or freak waves, high power lasers, and the propagation of large changes in pressure waves such as those that occur in shock waves and blast waves.For linear waves or small amplitude waves a substantial theory is available and, in many situations, exact solutions can be found. For nonlinear waves the available theory is less developed and there are relatively few nonlinear wave equations for which exact solutions are known. However, the so-called Inverse Scattering Transform method has been used to find solutions and understand important properties of the underlying equations and has led to a wide class of solutions of physically significant nonlinear wave equations. This method can be used to find so-called solitons solutions, stable localized waves which have special interaction properties and have been found to apply widely, for example in water waves, electromagnetic waves, lattice dynamics, magnetic waves, and elasticity. The investigator will consider extensions of this method to new classes of nonlinear equations, such as fractional nonlinear wave equations and their soliton solutions. Fractional equations themselves have numerous applications, including to the study of amorphous materials. Additionally, the method will be applied to study the properties of a class of multidimensional solitons, that is lump-type waves that vanish in all directions, which are relevant also for equations that arise in quantum mechanics and optics.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

波浪运动是自然界的基础：光波影响人们看东西的方式，声波使人们听到声音，水波对船只和潮汐能的运动至关重要。本项目侧重于非线性波传播，即对大振幅或高功率信号的研究，旨在提高科学家对大振幅波和波能传递相关动力学的洞察力和理解能力。研究结果将有利于非线性波动运动的应用，包括海啸，流氓或异常波，高功率激光，以及在冲击波和冲击波中发生的压力波的大变化的传播。对于线性波或小振幅波，有一个可靠的理论，在许多情况下，可以找到精确的解。对于非线性波动，可用的理论还不太发达，而且已知精确解的非线性波动方程相对较少。然而，所谓的逆散射变换方法已经被用来寻找解和理解底层方程的重要性质，并导致了一系列物理上重要的非线性波动方程的解。该方法可用于寻找所谓的孤子解，即具有特殊相互作用性质的稳定局域波，已被发现广泛应用于水波、电磁波、晶格动力学、电磁波和弹性等领域。研究者将考虑将此方法扩展到新的非线性方程，如分数阶非线性波动方程及其孤子解。分数方程本身有许多应用，包括对非晶材料的研究。此外，该方法将应用于研究一类多维孤子的性质，即在所有方向上消失的块状波，这也与量子力学和光学中出现的方程有关。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。